Dynamic proteome trade-offs regulate bacterial cell size and growth in fluctuating nutrient environments

Bacteria dynamically regulate cell size and growth to thrive in changing environments. While previous studies have characterized bacterial growth physiology at steady-state, a quantitative understanding of bacterial physiology in time-varying environments is lacking. Here we develop a quantitative theory connecting bacterial growth and division rates to proteome allocation in time-varying nutrient environments. In such environments, cell size and growth are regulated by trade-offs between prioritization of biomass accumulation or division, resulting in decoupling of single-cell growth rate from population growth rate. Specifically, bacteria transiently prioritize biomass accumulation over production of division machinery during nutrient upshifts, while prioritizing division over growth during downshifts. When subjected to pulsatile nutrient concentration, we find that bacteria exhibit a transient memory of previous metabolic states due to the slow dynamics of proteome reallocation. This allows for faster adaptation to previously seen environments and results in division control which is dependent on the time-profile of fluctuations.

. Inclusion of degradation rate, µ X , is necessary to produce experimentally-observed size control overshoots. Generation-averaged dynamics of added volume (A), interdivision time (B), and cell volume ratio (C) from 400 single-cell volume trajectories with (dotted) and without (solid) inclusion of the division protein degradation rate, µ X . Increasing the degradation rate increases the amplitude of the size control overshoot.  pulse-length τ starve , the time required following downshift for the growth rate to return to within 99% of the pre-shift level was measured, given by τ recovery . B Quantification of the relationship of τ starve and τ recovery from the simulations in A, along with the analytical approximation for comparison. Parameters are identical to those used in Figure 5 of the main text (given in Table 1), except κ 0 n,high = 30 h −1 .   Figure 6. Dynamic resource allocation during exit from stationary phase. Single-cell simulation dynamics of amino acid mass fraction and division protein allocation fraction for E. coli experiencing pulses of nutrients with delay τ pulse starting from stationary phase. An increase in available nutrients results in an increase in the intracellular amino acid mass fraction. In response to the influx of resources, our model predicts that bacteria transiently prioritize ribosome production ( f R ) over division ( f X ) immediately following pulse exposure, similar to nutrient upshift behavior predicted in exponential phase. Parameters are identical to those used in Figure 5 of the main text (given in Table 1).   Table 1.

SUPPLEMENTARY NOTE 1: GROWTH RATE MAXIMIZATION TO OBTAIN RIBOSOMAL ALLOCATION FRACTION
During steady-state exponential growth, the rate of amino acid supply is balanced by the rate of amino acid consumption through protein synthesis to ensure that there is no net change in the amino acid concentration.
Furthermore, the rate of protein synthesis equals the rate of bacterial growth, so the cell is faced with the dual objectives of balancing and maximizing the amino acid flux in order to maximize the growth rate ( Supplementary Figure 7a,b). For a given translational efficiency κ t (a) and the nutritional efficiency κ n (a) (as determined by the growth medium), the organism must choose the ribosomal protein fraction f R (a) that balances the amino acid flux. This is implemented as follows [5]. At steady-state φ R = f R (Eq. (10)). Using the condition of steady-state in Eqs. (10) and (11) .
The steady-state growth rate is given by .  12) and (13)), and so can adjust quickly in response to changes in nutrient conditions. Following a nutrient shift, there is a sudden jump in a caused by a temporary mismatch in fluxes. The regulatory functions ensure that the fluxes quickly rebalance, eliminating any subsequent dramatic changes in a until the next change in c. The cell achieves further changes in flux by adjusting its ribosome mass fraction, φ R , via adjustment to the fraction of ribosomes allocated to synthesize additional ribosomes, f R . Importantly, altering φ R via f R is accomplished through protein synthesis and degradation, and thus occurs on a much slower timescale than changes to κ t and κ n . This behavior can be seen graphically using phase-plane analysis, in which the model trajectory first moves to the a nullcline by changing a without changing φ R , at which point the fluxes have been rebalanced, before then moving to the fixed point by changing both a and φ R (Supplementary Figure 7c,d).

PROTEOMICS DATA
In order to obtain a value for allocation fraction f X , the identity of the specific cell division proteins, collectively referred to as X proteins, must be known. Although multiple proteins may be involved in setting division timing, experimental evidence suggests that FtsZ is the main determinant of cell division control in E. coli [6,7]. As such, in order to obtain approximate values for f X we assume that X protein abundance is made up entirely of FtsZ proteins. Obtaining an estimate for f X requires that the parameters α and β be known, given that f X = α(φ max R − f R ) + β . In order to estimate these parameters, Eq. (S.13) can be fit to experimental data, where there are three fitting parameters, γα, γβ , and κ t (∆φ can be inferred from experimental data [8]), where γ = ρ c /X 0 m X . To find α and β explicitly, the value of γ must be known. To estimate its value, we assume that the division threshold, X 0 , is determined entirely by FtsZ, such that X 0 = M 0 X /m X , where M 0 X is the total mass of FtsZ at division, and m X is the mass of a single FtsZ protein. As a result, we obtain the expression γ = ρ c /M 0 X . Using proteomics data [9], we estimate that M 0 X ≈ 6 * 10 −4 pg. With the typical protein mass density of an E. coli cell given by ρ c ≈ 0.15 pg/µm 3 [10], the parameters α and β can be calculated using γ ≈ 250 µm −3 . Interestingly, the range of proteome fractions obtained from our calculated size control parameters α and β are similar to the measured range of proteome fractions for FtsZ, namely ∼ 10 −3 [1]. This further supports the notion that FtsZ is indeed responsible for division timing in E. coli.

SUPPLEMENTARY NOTE 4: GROWTH-RATE DEPENDENT CONTROL OF CELL SIZE
The coupled equations (8), (10), (11), and (17) in the Methods define the dynamics of the system. As X represents the accumulated number of division proteins, the amount of division molecules is reset to zero at cell birth (i.e.X(0) = 0). Thus, at steady state growth Eqs. (8) and (17) can be solved to obtaiñ where V 0 is cell volume at birth. If cells divide symmetrically (i.e. V d = 2V 0 ) at t = τ after accumulating the required number of X proteins such that X(τ) = X 0 or equivalentlyX(τ) = 1, cell size at division can be related to κ and k P , yielding where the cell volume at division is V d = V 0 e κτ . In the limit κ >> µ X , we arrive at where ∆ = V d − V 0 is the added volume per generation. Because k P and κ are both constant for a given growth medium, cells exhibit an adder mechanism in which a constant volume is added each generation regardless of birth size. In the opposite limit, in which κ << µ X , we get In slow-growing media, cells divide at a constant volume, and thus break from the adder mechanism and exhibit sizer behavior, in which cells divide at a set volume, regardless of birth volume.
Equation (S.8) can be rearranged to give the birth size as a function growth rate, such that As k P is also a function of growth rate, Eq. (27) can be used to modify the equation above so that the dependency of V 0 on κ is fully realized. As a result we obtain .

(S.12)
Outside of slow growing conditions, the effects of protein degradation are negligible. Assuming κ µ ns and κ µ X , the above equation simplifies to important to note that this theoretical maximum is nonphysical as it assumes that f X = φ X = 0, which is never the case given our definition of f X . The actually maximum growth rate occurs when φ R = φ max R , thus giving an upper limit to physical growth rate at κ max = κ t ∆φ . Eq. (S.13) also predicts that there is no maximum cell size. However, our expression for f X constrains cell size to a finite value. When allocation to ribosomes is maximal, φ X = β , such that the maximum birth volume V 0 is given by V max 0 = 1/γβ .

SUPPLEMENTARY NOTE 5: ALTERNATIVE ALLOCATION STRATEGIES
To further probe how the specific underlying resource allocation strategy impacts size and division control, two alternative strategies were simulated for f X . First a constant allocation strategy was simulated, in which f X in invariant to nutrient perturbations. The degradation rate, µ X , was left as a fitting parameter to ensure that our previous choice of µ X was not responsible for disagreements with data, and the value of f X was calculated such that the model would reproduce the initial steady-state added volume and interdivision time measurements. This yielded the parameters γ f const In addition, an allocation strategy in which f X is proportional to f R was also simulated. Specifically, f R X = α f R + β , where our previously calculated value for β was reused and α was calculated to reproduce the initial steady-state added volume and interdivision time measurements. µ X was again left as a fitting parameter, yielding γα = 0.656, γβ = 0.656, and µ X = 0. As can be seen in Figures 3b-d in the main text, this allocation strategy fails to predict the overshoot in interdivision time following upshift, and incorrectly predicts a decrease in added volume with an increase in nutrient quality. This behavior again can be understood from Eq. (S.9), where now k P (κ) = γ f R X κ, yielding ∆(κ) = V 0 (κ) = 1/(γα(κ/κ t + φ min R ) + γβ ) at steady-state. Thus, this allocation strategy predicts a decrease in cell size with increasing growth rate.

SUPPLEMENTARY NOTE 6: ANALYTICAL APPROXIMATION OF RELATIONSHIP BETWEEN STARVATION TIME AND GROWTH RATE RECOVERY TIME
Here we derive an analytical approximation relating the starvation time, τ starve , to the time required to fully recovery the pre-starvation growth rate, τ recovery . Specifically, we consider a scenario in which cells growing at steady-state experience sudden starvation of duration τ starve , followed by growth rescue via sudden nutrient exposure of identical quality to the initial growth conditions (Supplementary Figure 4a).
We start by remembering the coupled equations which govern bacterial growth rate in time-varying nutrient environments, where the growth rate is given by To understand how the starvation time is related to recovery time, we must first understand how recovery time is affected by the macromolecular composition of the cell during starvation. If growth is rescued in starved cells by a step-wise increase in extracellular nutrients, there is an immediate influx of amino acids (a) which drives both translational efficiency and ribosomal allocation fraction close to their maximal values, such that κ t (a) ≈ κ 0 t and f R (a) ≈ φ max R . Furthermore, Eq. (S.15) can be modified to express the dynamics in terms of the active ribosome fraction, φ act Thus immediately after upshift at t = 0, Eq. (S.15) is uncoupled from Eq. (S.14) and a closed-form expression can be obtained. Specifically, , (S.17) where φ p R is the active ribosome fraction in the nutrient-poor condition immediately before upshift, and φ r R is the active ribosome fraction in the nutrient-rich condition at steady-state. Although φ r R is a steady-state quantity which remains the same before and after starvation if the nutrient quality is identical, φ p R is dependent on the starvation time, τ starve . Similar to rescue, if the onset of starvation is step-wise such that κ n (0) = 0, then immediately following downshift the dynamics of a are da/dt = µ ns − κ t (a)φ act R . The regulatory functions of κ t (a) and κ n (a) ensure that the translational and nutritional fluxes are quickly balanced following nutrient perturbations. Consequently, the relaxation progresses along the a nullcline (Supplemen- The growth rate can be approximated as κ r ≈ κ 0 t φ r R − µ ns in the nutrient-rich environment, allowing the recovery time to be related to the pre(post)-starvation growth rate, τ recovery = 1 κ 0 t ∆φ ln κ 0 t ∆φ e µ ns τ starve − κ r − µ ns κ 0 t ∆φ − κ r − µ ns .

(S.24)
On physiologically relevant timescales, this analytical approximation explains the simulated behavior well (Supplementary Figure 4b).